Analytic and Geometric Methods in Inverse Problems and Imaging

反问题和成像中的解析和几何方法

• 批准号: RGPIN-2016-06329
• 负责人: Nachman, Adrian
• 金额: $ 1.97万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710251
• 关键词: Analytic; Geometric; Methods; Inverse; Problems

This proposal seeks to answer some fundamental mathematical questions in the fields of Inverse Problems and Image Processing, as well as to contribute specific applications to Medical Imaging. The field of Inverse Problems studies novel methods to obtain images from noninvasive measurements, for next generation imaging modalities. Image Processing treats images obtained from existing modalities and the mathematics involved in denoising, segmenting, registering and extracting information from them. Classically, the data in Inverse Problems consists of measurements in the exterior of the object under investigation, or on its boundary. There have been considerable advances by numerous researchers in the systematic study of such problems, but several challenging questions remain open, and new ideas for addressing some of them will be sought. In addition, a new class of inverse problems considers situations where some interior information can be obtained (for instance using Magnetic Resonance Imagers). In joint work with A. Tamasan and A. Timonov, we have found a connection between one such problem arising in electric impedance imaging and the theory of minimal surfaces in non-euclidean geometries (determined by the measured data) and of related weighted least gradient problems. In recent joint work with A. Tamasan and J. Veras, the practical application has lead us to an interesting novel boundary value problem which has not been previously considered in geometric measure theory and will be investigated. We will also seek to obtain corresponding efficient and scalable numerical algorithms for its solution, and apply these to experimental data obtained in collaboration with M.Joy's group at the University of Toronto. The study of weighted least gradient problems has also led us to several analytic and geometric questions in Image Analysis. In a seminal paper, Tadmor, Nessar and Vese introduced a multiscale decomposition of images based on a sequence of variational problems involving a similar regularization functional; they showed that this can be considered as a nonlinear harmonic decomposition, with successive terms adding finer scale details. In joint work with K. Modin and L. Rondi we have recently extended this approach to registration problems, thus obtaining a (multiplicative) harmonic decomposition of diffeomorphisms. Theoretical and practical implications of this novel decomposition will be investigated. We will also apply the multiscale approach to the inverse problem described above, as a possible tool for handling non-smooth impedances. Solution of some of the problems we plan to study is expected to have a significant impact in the development of next generation imaging modalities as well as to the image analysis of clinical data.

该提案旨在回答反问题和图像处理领域的一些基本数学问题，并为医学成像提供具体的应用。反问题领域研究从无创测量中获取图像的新方法，用于下一代成像模式。图像处理处理从现有模式获得的图像以及去噪、分割、注册和从中提取信息所涉及的数学。 传统上，反问题中的数据由所研究对象的外部或其边界的测量组成。许多研究人员在系统研究此类问题方面取得了相当大的进展，但一些具有挑战性的问题仍然悬而未决，并且将寻求解决其中一些问题的新想法。此外，一类新的反问题考虑了可以获得一些内部信息的情况（例如使用磁共振成像仪）。在与 A. Tamasan 和 A. Timonov 的合作中，我们发现了电阻抗成像中出现的此类问题与非欧几里得几何中的最小表面理论（由测量数据确定）和相关加权最小梯度理论之间的联系问题。在最近与 A. Tamasan 和 J. Veras 的合作中，实际应用使我们发现了一个有趣的新颖边值问题，该问题以前在几何测度理论中未曾考虑过，将进行研究。我们还将寻求为其解决方案获得相应的高效且可扩展的数值算法，并将其应用到与多伦多大学 M.Joy 团队合作获得的实验数据中。 加权最小梯度问题的研究也给我们带来了图像分析中的几个解析和几何问题。在一篇开创性的论文中，Tadmor、Nessar 和 Vese 介绍了基于涉及类似正则化函数的一系列变分问题的图像多尺度分解；他们表明，这可以被视为非线性谐波分解，连续项添加更精细的尺度细节。在与 K. Modin 和 L. Rondi 的合作中，我们最近将这种方法扩展到配准问题，从而获得微分同胚的（乘法）调和分解。我们将研究这种新颖分解的理论和实践意义。我们还将应用多尺度方法来解决上述逆问题，作为处理非平滑阻抗的可能工具。 我们计划研究的一些问题的解决预计将对下一代成像模式的开发以及临床数据的图像分析产生重大影响。